Heat oven to 375°F. Lightly grease a 9x13-inch baking dish.
Melt butter in a large saucepan over medium heat. Soften onion for 4–5 minutes; stir in garlic for 30 seconds.
Sprinkle in flour and cook, stirring, for 1 minute to form a blond roux.
Whisk in warm broth gradually. Simmer, whisking, until slightly thickened, 3–4 minutes.
Remove from heat. Stir in sour cream, green chiles, cumin, chili powder, salt, pepper, and lime juice.
Toss shredded chicken with about 1 cup of the white sauce; set aside.
Spread 1/2 cup sauce in the dish. Layer half the tortilla strips, half the chicken, half the corn and beans, and one-third of the cheeses. Spoon over more sauce.
Repeat layers with remaining tortillas, chicken, corn, and beans. Top with remaining sauce and the rest of the cheeses.
Cover with foil and bake for 20 minutes. Uncover and bake 10–15 minutes more until bubbling and lightly browned.
Rest 10 minutes before cutting. Sprinkle with cilantro and serve.